On symplectic resolutions and factoriality of Hamiltonian reductions

TL;DR

This paper investigates the properties of symplectic singularities arising from Hamiltonian reductions of 3-large representations, establishing conditions for factoriality and the existence of symplectic resolutions.

Contribution

It proves that these singularities are always terminal, characterizes when they are -factorial or locally factorial, and shows they lack symplectic resolutions in the semi-simple case.

Findings

Singularities are always terminal.

-factorial iff G has finite abelianization.

No symplectic resolutions for semi-simple G.

Abstract

Recently, Herbig--Schwarz--Seaton have shown that $3$-large representations of a reductive group $G$ give rise to a large class of symplectic singularities via Hamiltonian reduction. We show that these singularities are always terminal. We show that they are $\mathbb{Q}$-factorial if and only if $G$ has finite abelianization. When $G$ is connected and semi-simple, we show they are actually locally factorial. As a consequence, the symplectic singularities do not admit symplectic resolutions when $G$ is semi-simple. We end with some open questions.